HyperLogLog in Practice: Algorithmic Engineering of a
State of The Art Cardinality Estimation Algorithm
Stefan Heule
Marc Nunkesser
ETH Zurich and Google, Inc.
Google, Inc.
Google, Inc.
[email protected]
marcnunkesser
@google.com
[email protected]
ABSTRACT
Cardinality estimation has a wide range of applications and
is of particular importance in database systems. Various
algorithms have been proposed in the past, and the HyperLogLog algorithm is one of them. In this paper, we
present a series of improvements to this algorithm that reduce its memory requirements and significantly increase its
accuracy for an important range of cardinalities. We have
implemented our proposed algorithm for a system at Google
and evaluated it empirically, comparing it to the original
HyperLogLog algorithm. Like HyperLogLog, our improved algorithm parallelizes perfectly and computes the
cardinality estimate in a single pass.
1.
INTRODUCTION
Cardinality estimation is the task of determining the number
of distinct elements in a data stream. While the cardinality
can be easily computed using space linear in the cardinality,
for many applications, this is completely impractical and requires too much memory. Therefore, many algorithms that
approximate the cardinality while using less resources have
been developed. These algorithms play an important role in
network monitoring systems, data mining applications, as
well as database systems.
At Google, various data analysis systems such as Sawzall [15],
Dremel [13] and PowerDrill [9] estimate the cardinality of
very large data sets every day, for example to determine the
number of distinct search queries on google.com over a time
period. Such queries represent a hard challenge in terms of
computational resources, and memory in particular: For the
PowerDrill system, a non-negligible fraction of queries historically could not be computed because they exceeded the
available memory.
In this paper we present a series of improvements to the
HyperLogLog algorithm by Flajolet et. al. [7] that estimates cardinalities efficiently. Our improvements decrease
memory usage as well as increase the accuracy of the es-
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timate significantly for a range of important cardinalities.
We evaluate all improvements empirically and compare with
the HyperLogLog algorithm from [7]. Our changes to the
algorithm are generally applicable and not specific to our
system. Like HyperLogLog, our proposed improved algorithm parallelizes perfectly and computes the cardinality
estimate in a single pass.
Outline. The remainder of this paper is organized as follows; we first justify our algorithm choice and summarize
related work in Section 2. In Section 3 we give background
information on our practical use cases at Google and list
the requirements for a cardinality estimation algorithm in
this context. In Section 4 we present the HyperLogLog
algorithm from [7] that is the basis of our improved algorithm. In Section 5 we describe the improvements we made
to the algorithm, as well as evaluate each of them empirically. Section 6 explains advantages of HyperLogLog for
dictionary encodings in column stores. Finally, we conclude
in Section 7.
2.
RELATED WORK AND ALGORITHM
CHOICE
Starting with work by Flajolet and Martin [6] a lot of research has been devoted to the cardinality estimation problem. See [3, 14] for an overview and a categorization of available algorithms. There are two popular models to describe
and analyse cardinality estimation algorithms: Firstly, the
data streaming (ε, δ)-model [10, 11] analyses the necessary
space to get a (1 ± ε)-approximation with a fixed success
probability of δ, for example δ = 2/3, for cardinalities in
{1, . . . , n}. Secondly it is possible to analyse the relative
accuracy defined as the standard error of the estimator [3].
The algorithm previously implemented at Google [13, 15, 9]
was MinCount, which is presented
as algorithm one in [2].
√
It has an accuracy of 1.0/ m, where m is the maximum
number of hash values maintained (and thus linear in the
required memory). In the (ε, δ)-model, this algorithm needs
O(ε−2 log n) space, and it is near exact for cardinalities up to
m (modulo hash collisions). See [8] for the statistical analysis and suggestions for making the algorithm more computationally efficient.
The algorithm presented by Kane et al. [11] meets the lower
bound on space of Ω(e−2 +log n) in the (ε, δ)-model [10] and
is optimal in that sense. However, the algorithm is complex
and an actual implementation and its maintenance seems
out of reach in a practical system.
In [1], the authors compare six cardinality estimation algorithms including MinCount and LogLog [4] combined
with LinearCounting [16] for small cardinalities. The latter algorithm comes out as the winner in that comparison.
In [14] the authors compare 12 algorithms analytically and
the most promising 8 experimentally on a single data set
with 1.9 · 106 distinct elements, among them LogLog, LinearCounting and MultiresolutionBitmap [5], a multiscale version of LinearCounting. LinearCounting is
recommended as the algorithm of choice for the tested cardinality. LogLog is shown to have a better accuracy than all
other algorithms except LinearCounting and MultiresolutionBitmap on their input data. We are interested in
estimating multisets of much larger cardinalities well beyond 109 , for which LinearCounting is no longer attractive, as it requires too much memory for an accurate estimate. MultiresolutionBitmap has similar problems and
needs O(ε−2 log n) space in the (ε, δ)-model, which is growing faster than the memory usage of LogLog. The authors
of the study also had problems to run MultiresolutionBitmap with a fixed given amount of memory.
As in most database systems, a user can count the number
of distinct elements in a data set by issuing a count distinct
query. In many cases, such a query will be grouped by a
field (e.g., country, or the minute of the day) and counts the
distinct elements of another field (e.g., query-text of Google
searches). For this reason, a single query can lead to many
count distinct computations being carried out in parallel.
On an average day, PowerDrill performs about 5 million
such count distinct computations. As many as 99% of these
computations yield a result of 100 or less. This can be explained partly by the fact that some groups in a group-by
query can have only few values, which translates necessarily
into a small cardinality. On the other extreme, about 100
computations a day yield a result greater than 1 billion.
As to the precision, while most users will be happy with a
good estimate, there are some critical use cases where a very
high accuracy is required. In particular, our users value the
property of the previously implemented MinCount algorithm [2] that can provide near exact results up to a threshold and becomes approximate beyond that threshold.
Therefore, the key requirements for a cardinality estimation
algorithm can be summarized as follows:
HyperLogLog has been proposed by Flajolet et. al. [7] and
is an improvement of LogLog. It has been published after
√
the afore-mentioned studies. Its relative error is 1.04/ m
and it needs O(ε−2 log log n+log n) space in the (ε, δ)-model,
where m is the number of counters (usually less than one
byte in size). HyperLogLog is shown to be near optimal
among algorithms that are based on order statistics. Its
theoretical properties certify that it has a superior accuracy
for a given fixed amount of memory over MinCount and
many other practical algorithms. The fact that LogLog
(with LinearCounting for small values cardinalities), on
which HyperLogLog improves, performed so well in the
previous experimental studies confirms our choice.
• Accuracy. For a fixed amount of memory, the algorithm should provide as accurate an estimate as possible. Especially for small cardinalities, the results
should be near exact.
• Memory efficiency. The algorithm should use the available memory efficiently and adapt its memory usage to
the cardinality. That is, the algorithm should use less
than the user-specified maximum amount of memory
if the cardinality to be estimated is very small.
• Estimate large cardinalities. Multisets with cardinalities well beyond 1 billion occur on a daily basis, and
it is important that such large cardinalities can be estimated with reasonable accuracy.
In [12], Lumbroso analyses an algorithm similar to HyperLogLog that uses the inverse of an arithmetic mean instead
of the harmonic mean as evaluation function. Similar to our
empirical bias correction described in Section 5.2, he performs a bias correction for his estimator that is based on a
full mathematical bias analysis of its “intermediate regime”.
3.
PRACTICAL REQUIREMENTS
FOR CARDINALITY ESTIMATION
In this section we present requirements for a cardinality estimation algorithm to be used in PowerDrill [9]. While we
were driven by our particular needs, many of these requirements are more general and apply equally to other applications.
PowerDrill is a column-oriented datastore as well as an interactive graphical user interface that sends SQL-like queries
to its backends. The column store uses highly memoryoptimized data structures to allow low latency queries over
datasets with hundreds of billions of rows. The system heavily relies on in-memory caching and to a lesser degree on the
type of queries produced by the frontend. Typical queries
group by one or more data fields and filter by various criteria.
• Practicality. The algorithm should be implementable
and maintainable.
4.
THE HYPERLOGLOG ALGORITHM
The HyperLogLog algorithm uses randomization to approximate the cardinality of a multiset. This randomization
is achieved by using a hash function h that is applied to every element that is to be counted. The algorithm observes
the maximum number of leading zeros that occur for all
hash values, where intuitively hash values with more leading zeros are less likely and indicate a larger cardinality. If
the bit pattern 0%−1 1 is observed at the beginning of a hash
value, then a good estimation of the size of the multiset is 2%
(assuming the hash function produces uniform hash values).
To reduce the large variability that such a single measurement has, a technique known as stochastic averaging [6] is
used. To that end, the input stream of data elements S is
divided into m substreams Si of roughly equal size, using
the first p bits of the hash values, where m = 2p . In each
substream, the maximum number of leading zeros (after the
initial p bits that are used to determine the substream) is
measured independently. These numbers are kept in an array of registers M , where M [i] stores the maximum number
of leading zeros plus one for substream with index i. That
is,
M [i] = max %(x)
x∈Si
where %(x) denotes the number of leading zeros in the binary representation of x plus one. Note that by convention
maxx∈∅ %(x) = −∞. Given these registers, the algorithm
then computes the cardinality estimate as the normalized
bias corrected harmonic mean of the estimations on the substreams as
2
E := αm · m ·
m
X
Require: Let h : D → {0, 1}32 hash data from domain D.
Let m = 2p with p ∈ [4..16].
Phase 0: Initialization.
1: Define α16 = 0.673, α32 = 0.697, α64 = 0.709,
2:
αm = 0.7213/(1 + 1.079/m) for m ≥ 128.
3: Initialize m registers M [0] to M [m − 1] to 0.
Phase 1: Aggregation.
4: for all v ∈ S do
5:
x := h(v)
6:
idx := hx31 , . . . , x32−p i2
{ First p bits of x }
7:
w := hx31−p , . . . , x0 i2
8:
M [idx] := max{M [idx], %(w)}
9: end for
Phase 2: Result computation.
!
2
2
10: E := αm m ·
j=1
Z
m
0
∞
log2
2+u
1+u
m
−1
du
Full details on the algorithm, as well as an analysis of its
properties can be found in [7]. In a practical setting, however, this algorithm has a series of problems, which Flajolet
et. al. address by presenting a practical variant of the algorithm. This second algorithm uses 32 bit hash values with
the precision argument p in the range [4..16]. The following modifications are applied to the algorithm. For a full
explanation of these changes, see [7].
1. Initialization of registers. The registers are initialized to 0 instead of −∞ to avoid the result 0 for
n m log m where n is the cardinality of the data
stream (i.e., the value we are trying to estimate).
2. Small range correction. Simulations by Flajolet et. al.
show that for n < 52 m nonlinear distortions appear
that need to be corrected. Thus, for this range LinearCounting [16] is used.
3. Large range corrections. When n starts to approach
232 ≈ 4 · 109 , hash collisions become more and more
likely (due to the 32 bit hash function). To account
for this, a correction is used.
The full practical algorithm is shown in Figure 1. In the
remainder of this paper, we refer to it as HllOrig .
5.
IMPROVEMENTS TO HYPERLOGLOG
In this section we propose a number of improvements to the
HyperLogLog algorithm. The improvements are presented
as a series of individual changes, and we assume for every
step that all previously presented improvements are kept.
We call the final algorithm HyperLogLog++ and show its
pseudo-code in Figure 6.
5.1
X
−M [j]
2
{ The “raw” estimate }
j=0
where
αm :=
!−1
m−1
−M[j]
Using a 64 Bit Hash Function
An algorithm that only uses the hash code of the input values is limited by the number of bits of the hash codes when it
comes to accurately estimating large cardinalities. In particular, a hash function of L bits can at most distinguish
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
if E ≤ 25 m then
Let V be the number of registers equal to 0.
if V 6= 0 then
E ∗ := LinearCounting(m, V )
else
E ∗ := E
end if
1 32
else if E ≤ 30
2 then
∗
E := E
else
E ∗ := −232 log(1 − E/232 )
end if
return E ∗
Define LinearCounting(m, V )
Returns the LinearCounting cardinality estimate.
24: return m log(m/V )
Figure 1: The practical variant of the HyperLogLog algorithm as presented in [7]. We use
LSB 0 bit numbering.
2L different values, and as the cardinality n approaches 2L ,
hash collisions become more and more likely and accurate
estimation gets impossible.
A useful property of the HyperLogLog algorithm is that
the memory requirement does not grow linearly with L, unlike other algorithms such as MinCount or LinearCounting. Instead, the memory requirement is determined by the
number of registers and the maximum size of %(w) (which
is stored in the registers). For a hash function of L bits and
a precision p, this maximum value is L + 1 − p. Thus, the
memory required for the registers is dlog2 (L + 1 − p)e · 2p
bits. The algorithm HllOrig uses 32 bit hash codes, which
requires 5 · 2p bits.
To fulfill the requirement of being able to estimate multisets of cardinalities beyond 1 billion, we use a 64 bit hash
function. This increases the size of a single register by only
a single bit, leading to a total memory requirement of 6 · 2p .
Only if the cardinality approaches 264 ≈ 1.8 · 1019 , hash collisions become a problem; we have not needed to estimate
inputs with a size close to this value so far.
With this change, the large range correction for cardinalities
close to 232 used in HllOrig is no longer needed. It would be
possible to introduce a similar correction if the cardinality
approaches 264 , but it seems unlikely that such cardinalities are encountered in practice. If such cardinalities occur,
however, it might make more sense to increase the number
of bits for the hash function further, especially given the low
additional cost in memory.
Estimating Small Cardinalities
The raw estimate of HllOrig (cf. Figure 1, line 10) has a
large error for small cardinalities. For instance, for n = 0
the algorithm always returns roughly 0.7m [7]. To achieve
better estimates for small cardinalities, HllOrig uses LinearCounting [16] below a threshold of 25 m and the raw
estimate above that threshold
In simulations, we noticed that most of the error of the raw
estimate is due to bias; the algorithm overestimates the real
cardinality for small sets. The bias is larger for smaller n,
e.g., for n = 0 we already mentioned that the bias is about
0.7m. The statistical variability of the estimate, however, is
small compared to the bias. Therefore, if we can correct for
the bias, we can hope to get a better estimate, in particular
for small cardinalities.
Experimental Setup. To measure the bias, we ran a version
of Hll64Bit that does not use LinearCounting and measured the estimate for a range of different cardinalities. The
HyperLogLog algorithm uses a hash function to randomize the input data, and will thus, for a fixed hash function
and input, return the same results. To get reliable data we
ran each experiment for a fixed cardinality and precision on
5000 different randomly generated data sets of that cardinality. Intuitively, the distribution of the input set should
be irrelevant as long as the hash function ensures an appropriate randomization. We were able to convince ourselves of
this by considering various data generation strategies that
produced differently distributed data and ensuring that our
results were comparable. We use this approach of computing
results on randomly generated datasets of the given cardinality for all experiments in this paper.
Note that the experiments need to be repeated for every possible precision. For brevity, and since the results are qualitatively similar, we illustrate the behavior of the algorithm
by considering only precision 14 here and in the remainder
of the paper.
We use the same proprietary 64 bit hash function for all
experiments. We have tested the algorithm with a variety of
hash functions including MD5, Sha1, Sha256, Murmur3,
as well as several proprietary hash functions. However, in
our experiments we were not able to find any evidence that
any of these hash functions performed significantly better
than others.
Empirical Bias Correction. To determine the bias, we
calculate the mean of all raw estimates for a given cardinality minus that cardinality. In Figure 2 we show the average raw estimate with 1% and 99% quantiles. We also
Algorihm
Hʟʟ 64Bɪᴛ
60000
Raw estimate
5.2
80000
40000
20000
0
0
20000
40000
Cardinality
60000
80000
Figure 2: The average raw estimate of the Hll64Bit
algorithm to illustrate the bias of this estimator for
p = 14, as well as the 1% and 99% quantiles on 5000
randomly generated data sets per cardinality. Note
that the quantiles and the median almost coincide
for small cardinalities; the bias clearly dominates
the variability in this range.
show the x = y line, which would be the expected value for
an unbiased estimator. Note that only if the bias accounts
for a significant fraction of the overall error can we expect
a reduced error by correcting for the bias. Our experiments
show that at the latest for n > 5m the correction does no
longer reduce the error significantly.
With this data, for any given cardinality we can compute
the observed bias and use it to correct the raw estimate.
As the algorithm does not know the cardinality, we record
for every cardinality the raw estimate as well as the bias so
that the algorithm can use the raw estimate to look up the
corresponding bias correction. To make this practical, we
choose 200 cardinalities as interpolation points, for which
we record the average raw estimate and bias. We use knearest neighbor interpolation to get the bias for a given
raw estimate (for k = 6)1 . In the pseudo-code in Figure 6
we use the procedure EstimateBias that performs the knearest neighbor interpolation.
Deciding Which Algorithm To Use. The procedure described so far gives rise to a new estimator for the cardinality,
namely the bias-corrected raw estimate. This procedure corrects for the bias using the empirically determined data for
cardinalities smaller than 5m and uses the unmodified raw
estimate otherwise. To evaluate how well the bias correction
1
The choice of k = 6 is rather arbitrary. The best value of k
could be determined experimentally, but we found that the
choice has only a minuscule influence.
works, and to decide if this algorithm should be used in favor of LinearCounting, we perform another experiment,
using the bias-corrected raw estimate, the raw estimate as
well as LinearCounting. We ran the three algorithms for
different cardinalities and compare the distribution of the
error. Note that we use a different dataset for this second
experiment to avoid overfitting.
For small cardinalities, LinearCounting is still better than
the bias-corrected raw estimate2 . Therefore, we determine
the intersection of the error curves of the bias-corrected raw
estimate and LinearCounting to be at 11500 for precision 14 and use LinearCounting to the left, and the biascorrected raw estimate to the right of that threshold.
As with the bias correction, the algorithm does not have
access to the true cardinality to decide on which side of
the threshold the cardinality lies, and thus which algorithm
should be used. However, again we can use one of the estimates to make the decision. Since the threshold is in a range
where LinearCounting has a fairly small error, we use its
estimate and compare it with the threshold. We call the resulting algorithm that combines the LinearCounting and
the bias-corrected raw estimate HllNoBias .
LɪɴᴇᴀʀCᴏᴜɴᴛɪɴɢ
Bias Corrected Raw Estimate of Hʟʟ 64Bɪᴛ
0.015
Median relative error
As shown in Figure 3, for cardinalities that up to about
61000, the bias-corrected raw estimate has a smaller error
than the raw estimate. For larger cardinalities, the error of
the two estimators converges to the same level (since the bias
gets smaller in this range), until the two error distributions
coincide for cardinalities above 5m.
Algorihm
Raw Estimate of Hʟʟ 64Bɪᴛ
0.010
0.005
0.000
raw estimate in combination with LinearCounting has a
series of advantages compared to combining the raw estimate and LinearCounting:
• The error for an important range of cardinalities is
smaller than the error of Hll64Bit . For precision 14,
this range is roughly between 18000 and 61000 (cf. Figure 3).
• The resulting algorithm does not have a significant
bias. This is not true for Hll64Bit (or HllOrig ), which
uses the raw estimate for cardinalities above the threshold of 5/2m. However, at that point, the raw estimate
is still significantly biased, as illustrated in Figure 4.
• Both algorithms use an empirically determined threshold to decide which of the two sub-algorithms to use.
However, the two relevant error curves for HllNoBias
are less steep at the threshold compared to Hll64Bit
(cf. Figure 3). This has the advantage that a small
error in the threshold has smaller consequences for the
accuracy of the resulting algorithm.
2
This is not entirely true, for very small cardinalities it
seems that the bias-corrected raw estimate has again a
smaller error, but a higher variability. Since LinearCounting also has low error, and depends less on empirical data,
we decided to use it for all cardinalities below the threshold.
20000
40000
Cardinality
60000
80000
Figure 3: The median error of the raw estimate,
the bias-corrected raw estimate, as well as LinearCounting for p = 14. Also shown are the 5%
and 95% quantiles of the error. The measurements
are based on 5000 data points per cardinality.
5.3
Advantages of Bias Correction. Using the bias-corrected
0
Sparse Representation
HllNoBias requires a constant amount of memory throughout the execution of 6m bits, regardless of n, violating our
memory efficiency requirement. If n m, then most of the
registers are never used and thus do not have to be represented in memory. Instead, we can use a sparse representation that stores pairs (idx, %(w)). If the list of such pairs
would require more memory than the dense representation
of the registers (i.e., 6m bits), the list can be converted to
the dense representation.
Note that pairs with the same index can be merged by keeping the one with the highest %(w) value. Various strategies
can be used to make insertions of new pairs into the sparse
representation as well as merging elements with the same index efficient. In our implementation we represent (idx, %(w))
pairs as a single integer by concatenating the bit patterns
for idx and %(w) (storing idx in the higher-order bits of the
integer).
Our implementation then maintains a sorted list of such integers. Furthermore, to enable quick insertion, a separate
set is kept where new elements can be added quickly without keeping them sorted. Periodically, this temporary set is
sorted and merged with the list (e.g., if it reaches 25% of the
maximum size of the sparse representation), removing any
pairs where another pair with the same index and a higher
%(w) value exists.
Because the index is stored in the high-order bits of the integer, the sorting ensures that pairs with the same index occur
consecutively in the sorted sequence, allowing the merge to
Algorihm
Hʟʟ 64Bɪᴛ
Hʟʟ NᴏBɪᴀs
0.03
1. idx0 consists of the p0 most significant bits of h(v), and
since p < p0 , we can determine idx by taking the p
most significant bits of idx0 .
0.02
Median relative bias
follows. Let h(v) be the hash value for the underlying data
element v.
2. For %(w) we need the number of leading zeros of the
bits of h(v) after the index bits, i.e., of bits 63 − p to
0. The bits 63 − p to 64 − p0 are known by looking
at idx0 . If at least one of these bits is one, then %(w)
can be computed using only those bits. Otherwise,
bits 63 − p to 64 − p0 are all zero, and using %(w0 ) we
know the number of leading zeros of the remaining bits.
Therefore, in this case we have %(w) = %(w0 ) + (p0 − p).
0.01
0.00
-0.01
0
20000
40000
Cardinality
60000
80000
Figure 4:
The median bias of HllOrig and
HllNoBias . The measurements are again based on
5000 data points per cardinality.
happen in a single linear pass over the sorted set and the
list. In the pseudo-code of Figure 6, this merging happens
in the subroutine Merge.
The computation of the overall result given a sparse representation in phase 2 of the algorithm can be done in a
straight forward manner by iterating over all entries in the
list (after the temporary set has been merged with the list),
and assuming that any index not present in the list has a
register value of 0. As we will explain in the next section,
this is not necessary in our final algorithm and thus is not
part of the pseudo-code in Figure 6.
The sparse representation reduces the memory consumption
for cases where the cardinality n is small, and only adds a
small runtime overhead by amortizing the cost of searching
and merging through the use of the temporary set.
5.3.1
Higher Precision for the Sparse Representation
Every item in the sparse representation requires p + 6 bits,
namely to store the index (p bits) and the value of that registers (6 bits). In the sparse representation we can choose
to perform all operations with a different precision argument p0 > p. This allows us to increase the accuracy in
cases where only the sparse representation is used (and it
is not necessary to convert to the normal representation).
If the sparse representation gets too large and reaches the
user-specified memory threshold of 6m bits, it is possible to
fall back to precision p and switch to the dense representation. Note that falling back from p0 to the lower precision p is always possible: Given a pair (idx0 , %(w0 )) that has
been determined with precision p0 , one can determine the
corresponding pair (idx, %(w)) for the smaller precision p as
This computation is done in DecodeHash of Figure 6. It is
possible to compute at a different, potentially much higher
accuracy p0 in the sparse representation, without exceeding
the memory limit indicated by the user through the precision parameter p. Note that choosing a suitable value for
p0 is a trade-off. The higher p0 is, the smaller the error for
cases where only the sparse representation is used. However,
at the same time as p0 gets larger, every pair requires more
memory which means the user-specified memory threshold
is reached sooner in the sparse representation and the algorithm needs to switch to the dense representation earlier.
Also note that one can increase p0 up to 64, at which points
the full hash code is kept.
We use the name HllSparse1 to refer to this algorithm. To
illustrate the increased accuracy, Figure 5 shows the error
distribution with and without the sparse representation.
5.3.2
Compressing the Sparse Representation
So far, we presented the sparse representation to use a temporary set and a list which is kept sorted. Since the temporary set is used for quickly adding new elements and merged
with the list before it gets large, using a simple implementation with some built-in integer type works well, even if
some bits per entry are wasted (due to the fact that builtin integer types may be too wide). For the list, however,
we can exploit two facts to store the elements more compactly. First of all, there is an upper limit on the number
of bits used per integer, namely p0 + 6. Using an integer of
fixed width (e.g., int or long as offered in many programming languages) might be wasteful. Furthermore, the list is
guaranteed to be sorted, which can be exploited as well.
We use a variable length encoding for integers that uses variable number of bytes to represent integers, depending on
their absolute value. Furthermore, we use a difference encoding, where we store the difference between successive elements in the list. That is, for a sorted list a1 , a2 , a3 , . . . we
would store a1 , a2 − a1 , a3 − a2 , . . .. The values in such a
list of differences have smaller absolute values, which makes
the variable length encoding even more efficient. Note that
when sequentially going through the list, the original items
can easily be recovered.
We use the name HllSparse2 if only the variable length encoding is used, and HllSparse3 if additionally the difference
0.015
Median relative error
Algorihm
Hʟʟ NᴏBɪᴀs (without sparse representation)
Hʟʟ++ (with sparse representation)
p
m
10
12
14
16
1024
4096
16384
65536
HllSparse1
HllSparse2
HllSparse3
192.00
768.00
3072.00
12288.00
316.45
1261.82
5043.91
20174.64
420.73
1962.18
8366.45
35616.73
Hll++
534.27
2407.73
12107.00
51452.64
0.010
Table 1: The maximum number of pairs with distinct index that can be stored before the representation reaches the size of the dense representation,
i.e., 6m bits. All measurements have been repeated
for different inputs, for p0 = 25.
0.005
and using one bit (e.g., the least significant bit) to indicate
whether it is present or not. We use the following encoding:
If the bits hx63−p , . . . , x64−p0 i are all 0, then the resulting
integer is
hx63 , . . . , x64−p0 i || h%(w0 )i || h1i
0.000
0
5000
10000
Cardinality
15000
20000
(where we use || to concatenate bits). Otherwise, the pair is
encoded as
hx63 , . . . , x64−p0 i || h0i
Figure 5: Comparison of HllNoBias and Hll++ to
illustrate the increased accuracy, here for p0 = 25,
with 5% and 95% quantiles. The measurements are
on 5000 data points per cardinality.
encoding is used.
Note that introducing these two compression steps is possible in an efficient way, as the sorted list of encoded hash
values is only updated in batches (when merging the entries
in the set with the list). Adding a single new value to the
compressed list would be expensive, as one has to potentially
read the whole list in order to even find the correct insertion
point (due to the difference encoding). However, when the
list is merged with the temporary set, then the list is traversed sequentially in any case. The pseudo-code in Figure 7
uses the not further specified subroutine DecodeSparse to
decompress the variable length and difference encoding in a
straight forward way.
5.3.3
Encoding Hash Values
It is possible to further improve the storage efficiency with
the following observations. If the sparse representation is
used for the complete aggregation phase, then in the result
computation HllNoBias will always use LinearCounting to
determine the result. This is because the maximum number
of hash values that can be stored in the sparse representation is small compared to the cardinality threshold of where
to switch from LinearCounting to the bias-corrected raw
estimate. Since LinearCounting only requires the number of distinct indices (and m), there is no need for %(w0 )
from the pair. The value %(w0 ) is only used when switching from the sparse to the normal representation, and even
then only if the bits hx63−p , . . . , x64−p0 i are all 0. For a good
hash function with uniform hash values, the value %(w0 ) only
0
needs to be stored with probability 2p−p .
This idea can be realized by only storing %(w0 ) if necessary,
The least significant bit allows to easily decode the integer
again. Procedures EncodeHash and DecodeHash of Figure 7 implement this encoding.
In our implementation3 we fix p0 = 25, as this provides
very high accuracy for the range of cardinalities where the
sparse representation is used. Furthermore, the 25 bits for
the index, 6 bits for %(w0 ) and one indicator bit fit nicely into
a 32 bit integer, which is useful from a practical standpoint.
We call this algorithm HyperLogLog++ (or Hll++ for short),
which is shown in Figure 6 and includes all improvements
from this paper.
5.3.4
Space Efficiency
In Table 1 we show the effects of the different encoding
strategies on the space efficiency of the sparse encoding for
a selection of precision parameters p. The less memory a
single pair requires on average, the longer can the algorithm
use the sparse representation without switching to the dense
representation. This directly translates to a high precision
for a larger range of cardinalities.
For instance, for precision 14, storing every element in the
sparse representation as an integer would require 32 bits.
The variable length encoding reduces this to an average of
19.49 bits per element. Additionally introducing a difference
encoding requires 11.75 bits per element and using the improved encoding of hash values further decreases this value
to 8.12 bits on average.
5.4
Evaluation of All Improvements
To evaluate the effect of all improvements, we ran HllOrig as
presented in [7] and Hll++. The error distribution clearly
illustrates the positive effects of our changes on the accuracy of the estimate. Again, a fresh dataset has been used
3
This holds for all backends except for our own column store,
where we use both p0 = 20 and p0 = 25, also see Section 6.
Input: The input data set S, the precision p, the precision
p0 used in the sparse representation where p ∈ [4..p0 ] and
p0 ≤ 64. Let h : D → {0, 1}64 hash data from domain D
to 64 bit values.
Phase 0: Initialization.
0
1: m := 2p ; m0 := 2p
2: α16 := 0.673; α32 := 0.697; α64 := 0.709
3: αm := 0.7213/(1 + 1.079/m) for m ≥ 128
4: format := sparse
5: tmp_set := ∅
6: sparse_list := []
Phase 1: Aggregation.
7: for all v ∈ S do
8:
x := h(v)
9:
switch format do
10:
case normal
11:
idx := hx63 , . . . , x64−p i2
12:
w := hx63−p , . . . , x0 i2
13:
M [idx] := max{M [idx], %(w)}
14:
end case
15:
case sparse
16:
k := EncodeHash(x, p, p0 )
17:
tmp_set := tmp_set ∪ {k}
18:
if tmp_set is too large then
19:
sparse_list :=
20:
Merge(sparse_list, Sort(tmp_set))
21:
tmp_set := ∅
22:
if |sparse_list| > m · 6 bits then
23:
format := normal
24:
M := ToNormal(sparse_list)
25:
end if
26:
end if
27:
end case
28:
end switch
29: end for
Phase 2: Result computation.
30: switch format do
31:
case sparse
32:
sparse_list := Merge(sparse_list, Sort(tmp_set))
33:
return LinearCounting(m0 , m0 −|sparse_list|)
34:
end case
35:
case normal
!
−1
m−1
36:
2
E := αm m ·
X
2
−M [j]
j=0
37:
E 0 := (E ≤ 5m) ? (E − EstimateBias(E, p)):E
38:
Let V be the number of registers equal to 0.
39:
if V 6= 0 then
40:
H := LinearCounting(m, V )
41:
else
42:
H := E 0
43:
end if
44:
if H ≤ Threshold(p) then
45:
return H
46:
else
47:
return E 0
48:
end if
49:
end case
50: end switch
Figure 6: The Hll++ algorithm that includes all the
improvements presented in this paper. Some auxiliary procedures are given in Figure 7.
Define LinearCounting(m, V )
Returns the LinearCounting cardinality estimate.
1: return m log(m/V )
Define Threshold(p)
Returns empirically determined threshold (we provide
the values from our implementation at http://goo.gl/
iU8Ig).
Define EstimateBias(E, p)
Returns the estimated bias, based on the interpolating
with the empirically determined values.
Define EncodeHash(x, p, p0 )
Encodes the hash code x as an integer.
2: if hx63−p , . . . , x64−p0 i = 0 then
3:
return hx63 , . . . , x64−p0 i || h%(hx63−p0 , . . . , x0 i)i || h1i
4: else
5:
return hx63 , . . . , x64−p0 i || h0i
6: end if
Define GetIndex(k, p)
Returns the index with precision p stored in k
7: if hk0 i = 1 then
8:
return hkp+6 , . . . , k6 i
9: else
10:
return hkp+1 , . . . , k1 i
11: end if
Define DecodeHash(k, p, p0 )
Returns the index and %(w) with precision p stored in k
12: if hk0 i = 1 then
13:
r := hk6 , . . . , k1 i + (p0 − p)
14: else
15:
r := %(hkp0 −p−1 , . . . , k1 i)
16: end if
17: return (GetIndex(k, p), r)
Define ToNormal(sparse_list, p, p0 )
Converts the sparse representation to the normal one
18: M := NewArray(m)
19: for all k ∈ DecodeSparse(sparse_list) do
20:
(idx, r) := DecodeHash(k, p, p0 )
21:
M [idx] := max{M [idx], %(w)}
22: end for
23: return M
Define Merge(a, b)
Expects two sorted lists a and b, where the first is compressed using a variable length and difference encoding.
Returns a list that is sorted and compressed in the same
way as a, and contains all elements from a and b, except
for entries where another element with the same index,
but higher %(w) value exists. This can be implemented
in a single linear pass over both lists.
Define DecodeSparse(a)
Expects a sorted list that is compressed using a variable
length and difference encoding, and returns the elements
from that list after it has been decompressed.
Figure 7: Auxiliary procedures for the Hll++ algorithm.
0.015
Algorihm
Hʟʟ Oʀɪɢ
Median relative error
Hʟʟ++
HyperLogLog has the useful property that not the full
hash code is required for the computation. Instead, it suffices to know the first p bits (or p0 if the sparse representation
with higher accuracy from Section 5.3 is used) as well as the
number of leading zeros of the remaining bits.
0.010
This leads to a smaller maximum number of distinct values
for the data column. While there are 264 possible distinct
hash values if the full hash values were to be stored, there
0
are only 2p · (64 − p0 − 1) + 2p · (2p −p − 1) different values
that our integer encoding from Section 5.3.3 can take4 . This
reduces the maximum possible size of the dictionary by a
major factor if p0 64
0.005
0.000
0
20000
40000
Cardinality
60000
80000
Figure 8: Comparison of HllOrig with Hll++. The
mean relative error as well as the 5% and 95% quantiles are shown. The measurements are on 5000 data
points per cardinality.
for this experiment, and the results of the comparison for
precision 14 are shown in Figure 8.
First of all, there is a spike in the error of HllOrig , almost
exactly at n = 5/2m = 40960. The reason for this is that
the ideal threshold of when to switch from LinearCounting to the raw estimate in HllOrig is not precisely at 5/2m.
As explained in Section 5.2, a relatively small error in this
threshold leads to a rather large error in the overall error,
because the error curve of the raw estimate is fairly steep.
Furthermore, even if the threshold was determined more precisely for HllOrig , its error would still be larger than that of
Hll++ in this range of cardinalities. Our HyperLogLog++
algorithm does not exhibit any such behavior.
The advantage of the sparse representation is clearly visible; for cardinalities smaller than about 12000, the error of
our final algorithm Hll++ is significantly smaller than for
HllOrig without a sparse representation.
6.
As explained in [9], most column stores use a dictionary encoding for the data columns that maps values to identifiers.
If there is a large number of distinct elements, the size of
the dictionary can easily dominate the memory needed for
a given count distinct query.
IMPLICATIONS FOR DICTIONARY
ENCODINGS OF COLUMN STORES
In this section we focus on a property of HyperLogLog
(and Hll++ in particular) that we were able to exploit in
our implementation for the column-store presented in [9].
Given the expression materialization strategy of that column store, any cardinality estimation algorithm that computes the hash value of an expression will have to add a
data column of these values to the store. The advantage
of this approach is that the hash values for any given expression only need to be computed once. Any subsequent
computation can use the precomputed data column.
For example, our column store already bounds the size of
the dictionary to 10 million (and thus there will never be
264 different hash values). Nonetheless, using the default
parameters p0 = 20 and p = 14, there can still be at most
1.74 million values, which directly translates to a memory
saving of more than 82% for the dictionary if all input values
are distinct.
7.
CONCLUSIONS
In this paper we presented a series of improvements to the
HyperLogLog algorithm. Most of these changes are orthogonal to each other and can thus be applied independently to fit the needs of a particular application.
The resulting algorithm HyperLogLog++ fulfills the requirements listed in Section 3. Compared to the practical
variant of HyperLogLog from [7], the accuracy is significantly better for large range of cardinalities and equally
good on the rest. For precision 14 (and p0 = 25), the sparse
representation allows the average error for cardinalities up
to roughly 12000 to be smaller by a factor of 4. For cardinalities between 12000 and 61000, the bias correction allows
for a lower error and avoids a spike in the error when switching between sub-algorithms due to less steep error curves.
The sparse representation also allows for a more adaptive
use of memory; if the cardinality n is much smaller than m,
then HyperLogLog++ requires significantly less memory.
This is of particular importance in PowerDrill where often
many count distinct computations are carried out in parallel for a single count distinct query. Finally, the use of 64
bit hash codes allows the algorithm to estimate cardinalities
well beyond 1 billion.
All of these changes can be implemented in a straight-forward
way and we have done so for the PowerDrill system. We provide a complete list of our empirically determined parameters at http://goo.gl/iU8Ig to allow easier reproduction
of our results.
This can be seen as follows: There are 2p (64 − p0 − 1) many
0
encoded values that store %(w0 ), and similarly 2p (2p −p − 1)
many without it.
4
8.
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